The wind beneath our wings

Classical Mechanics Level 5

Many people often ascribe the lift from an airplane wing as purely due to Bernoulli's equation and the pressure gradients on the bottom and top of the wing. One very common, but ultimately physically false idea that employs Bernoulli's principle is the equal transit time model. In this model, one hypothesizes that the air flowing over the top of a wing takes the same amount of time as the air flowing past the bottom. The top of the wing is curved, air therefore moves faster over it, there's a pressure difference between top and bottom and voila!, we have lift.

Let's calculate how much lift is generated by a simple wing shape in this model. A Boeing 777 airplane flies at 250 m/s. The cross section of the wing is as follows. The bottom of the wing is 7 m wide. The top of the wing is an arc of a circle and the maximum distance between the top and bottom of the wing (which occurs right in the middle) is 0.7 m. What is the pressure difference in Pascals between the bottom and top surfaces (i.e. \(p_b-p_t\)) of the wing at sea level using the equal transit time model?

Note that the lift due to this pressure difference is not enough. More lift is actually generated than this because the air on top actually travels faster than the equal time model predicts.

Details and assumptions

  • The density of air at sea level is \(1.2~kg/m^3\).

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